You might consider going to Ville d' Este and Hadrian's Villa using public transportation. Ville d' Este is particularly interesting since the garden area with numerous fountains are landscaped into a hillside.
Another idea would be Ostia Antica, which is like Pompeii without the history. It does provide two amenities missing from Pompeii: shade and a sea breeze. This is also reached by public transportation or train.
If you are up to a LONG day you might consider Pompeii, itself. We took a bus tour to the site and were there for three hours. Pompeii is immense. When we were there last in 2009 the visitor center in the site was closed and it can be hot, dry, and did I forget to mention hot.
The catacombs of San Calixto or the Pope's summer home in nearby Castel Gondolfo is another good choice.
buon viaggio,